At time \(t_{\rm wr}-\Delta t\)  the blue supergiant to Wolf-Rayet phase
begins to take place, controlled by several quantities, namely its
duration \(\Delta t\)  and the manner the surface stellar properties evolve,
determined by the quintuplet (\(\dot{M}\) ,\(v_{\rm w}\) ,\(v_{\rm rot}\) ,
\(B_{\star }\) ,\(R_{\star }\) ).
The stellar wind velocity is modulated following [1]},
\(v_{\rm w}(r_{\rm in},t) = \Bigg \lbrace \begin{array}{lll}v_{\rm w}^{\rm bsg} / \sqrt{ f(\theta ) } & \mathrm {if} & t \le t_{\rm wr}-\Delta t , \\v_{\rm w}^{\rm bsg} / \sqrt{ f(\theta ) }+\Delta vF & \mathrm {if} & t_{\rm wr}-\Delta t < t and t \le t_{\rm wr}, \\v_{\rm w}^{\rm wr} & \mathrm {if} & t_{\rm wr} < t, \\\end{array}\) 
